Braiding statistics and classification of two-dimensional charge-2m superconductors

We study braiding statistics between quasiparticles and vortices in two-dimensional charge-2m (in units of e ) superconductors that are coupled to a Z_2m dynamical gauge field, where m is any positive integer. We show that there exist 16m types of braiding statistics when m is odd, but only 4m types when m is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-2m superconductors - or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with Z^f_2m symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with Z₄^f symmetry.